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Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]3 Marks
Question
Prove that $\cot ^{-1}(7)+2 \cot ^{-1}(3)=\frac{\pi}{4}$
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Answer

$\text { L.H.S. }=\cot ^{-1}(7)+2 \cot ^{-1}(3)$
$=\cot ^{-1}(7)+\cot ^{-1}(3)+\cot ^{-1}(3)$
$=\frac{\pi}{2}-\tan ^{-1}(7)+\frac{\pi}{2}-\tan ^{-1}(3)+\frac{\pi}{2}-\tan ^{-1}(3) \quad \ldots \ldots . .\left[\because \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\right]$
$=\frac{3 \pi}{2}-\left[\pi+\tan ^{-1}\left(\frac{7+3}{1-7 \times 3}\right)+\tan ^{-1}(3)\right] \ldots \ldots . .$
${\left[\because \tan ^{-1} x+\tan ^{-1} y=\pi+\tan ^{-1} \frac{x+y}{1-x y}, \text { if } x, y>0 \text { and } x y>1\right]}$
$=\frac{3 \pi}{2}-\pi-\left[\tan ^{-1}\left(\frac{10}{-20}\right)+\tan ^{-1}(3)\right]$
$=\frac{\pi}{2}-\left[\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}(3)\right]$
$=\frac{\pi}{2}-\left[\tan ^{-1}(3)-\tan ^{-1}\left(\frac{1}{2}\right)\right] \ldots \ldots . .\left[\because \tan ^{-1}(-x)=-\tan ^{-1}(x)\right]$
$=\frac{\pi}{2}-\left[\tan ^{-1}\left(\frac{3-\frac{1}{2}}{1+(3)\left(\frac{1}{2}\right)}\right)\right]$
$=\frac{\pi}{2}-\left[\tan ^{-1}\left(\frac{\frac{5}{2}}{\frac{5}{2}}\right)\right]$
$=\frac{\pi}{2}-\tan ^{-1}(1)$
$=\frac{\pi}{2}-\frac{\pi}{4}$
$=\frac{\pi}{4}$
$=\text { R.H.S. }$

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